Chapter 14.7, Problem 54E

A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m² per day, the north and south walls at a rate of 8 units/m² per day, the floor at a rate of 1 unit/m² per day, and the roof at a rate of 5 units/m² per day. Each wall must be at least 30 m long, the height must be at least 4 m, and the volume must be exactly 4000 m³.

(a) Find and sketch the domain of the heat loss as a function of the lengths of the sides.

(b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the boundary of the domain.)

(c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls were removed?